Takagi factorization

This diploma thesis, Takagi factorization, presents the reader a factorization of complex symmetric matrices. A complex symmetric matrix can not in general be diagonalizable but it can be showed, that it can always be unitary T- congruent to some diagonal matrix. That is called Takagi factorization. In the first chapter the basic properties of matrices and matrix diagonalizability are presented. A diagonalization is not always realisable in the case of a complex orthogonal matrix while for real symmetric and for complex Hermitian matrix the diagonalization always exists. An important theorem for understanding the process of diagonalization and similarity is Schur form. It is described and proved below. Short part of this dissertation is devoted to Japanese mathematician Teiji Takagi, followed by the main subject of this paper, discussion of Takagi factorization